1.1 Systems of Linear Equations

Notes

•1.1 Systems of Linear Equations A linear equation in variables x1, ..., xn can be written as a1x1 + a2x2 + ... + anxn = b, where the coefficients a1, ..., an and the constant b are real or complex numbers. The subscript n can be any positive integer. A system of linear equations (or linear system) is a collection of one or more such equations involving the same variables x1, ..., xn. A solution is a list (s1, s2, ..., sn) of numbers that makes each equation true when substituted for x1, ..., xn. The set of all solutions is called the solution set. Two systems are equivalent if they share the same solution set. A linear system has either no solution, exactly one solution, or infinitely many solutions. It is called consistent if it has at least one solution; otherwise, it is inconsistent. The key information of a linear system can be compactly recorded in a matrix. The coefficient matrix contains the coefficients aligned in columns. An augmented matrix adds a column with the constants from the right sides of the equations. The size of a matrix, m x n, indicates m rows and n columns. Elementary row operations include replacing a row by the sum of itself and a multipl...